Details
| Original language | English |
|---|---|
| Pages (from-to) | 436-452 |
| Number of pages | 17 |
| Journal | Journal of Geometry and Physics |
| Volume | 61 |
| Issue number | 2 |
| Publication status | Published - 1 Feb 2011 |
Abstract
We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S1) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.
Keywords
- Camassa-Holm equation, Degasperis-Procesi equation, Geodesic flow, Sectional curvature, Semidirect product
ASJC Scopus subject areas
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- General Physics and Astronomy
- Mathematics(all)
- Geometry and Topology
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In: Journal of Geometry and Physics, Vol. 61, No. 2, 01.02.2011, p. 436-452.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations
AU - Escher, Joachim
AU - Kohlmann, Martin
AU - Lenells, Jonatan
PY - 2011/2/1
Y1 - 2011/2/1
N2 - We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S1) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.
AB - We use geometric methods to study two natural two-component generalizations of the periodic Camassa-Holm and Degasperis-Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S1) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa-Holm equation, giving explicit examples of large subspaces of positive curvature.
KW - Camassa-Holm equation
KW - Degasperis-Procesi equation
KW - Geodesic flow
KW - Sectional curvature
KW - Semidirect product
UR - http://www.scopus.com/inward/record.url?scp=78349295721&partnerID=8YFLogxK
U2 - 10.1016/j.geomphys.2010.10.011
DO - 10.1016/j.geomphys.2010.10.011
M3 - Article
AN - SCOPUS:78349295721
VL - 61
SP - 436
EP - 452
JO - Journal of Geometry and Physics
JF - Journal of Geometry and Physics
SN - 0393-0440
IS - 2
ER -